Diophantine approximations related to rational values of G-functions
Makoto Nagata (2003)
Acta Arithmetica
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Makoto Nagata (2003)
Acta Arithmetica
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Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)
Colloquium Mathematicae
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We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions. ...
Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)
Journal de Théorie des Nombres de Bordeaux
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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
Yasutsugu Fujita (2007)
Acta Arithmetica
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Henk, Martin, Weismantel, Robert (2000)
Beiträge zur Algebra und Geometrie
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H. Kleiman (1976)
Journal für die reine und angewandte Mathematik
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Yang Hai, P. G. Walsh (2010)
Acta Arithmetica
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W. J. Ellison (1970-1971)
Séminaire de théorie des nombres de Bordeaux
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Alan Filipin (2009)
Acta Arithmetica
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Shin-ichi Katayama, Claude Levesque (2003)
Acta Arithmetica
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Ernst, Bruno (1996)
General Mathematics
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Jianhua Chen (2001)
Acta Arithmetica
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Umberto Zannier (2003)
Acta Arithmetica
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Pingzhi Yuan, Jiagui Luo (2010)
Acta Arithmetica
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Katalin Gyarmati (2001)
Acta Arithmetica
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Ayla Gafni (2014)
Acta Arithmetica
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We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².
Susil Kumar Jena (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.
Pingzhi Yuan (2004)
Acta Arithmetica
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